Preface	8
	Motivation and Purpose	8
	Highlights	9
	Intended Audience	10
	Acknowledgments	10
	Contents	12
	Notation	15
	Scalars, Vectors, and Tuples	15
	Vector Components and Compound Vectors	15
	Screws	16
	Matrices	16
	Sets	16
	Mechanism Symbols	17
	Maps	17
	Other Symbols	18
	List of Figures	19
	1 Introduction	22
	1.1 Historical Context	24
	1.2 Assumptions and Scope	26
	1.3 Reader's Guide	29
	References	30
	2 Singularity Types	34
	2.1 Forward and Inverse Singularities	34
	2.2 A Geometric Interpretation of Singularities	38
	2.2.1 Singularities Yield Shaky Mechanisms	38
	2.2.2 C-Space, Input, and Output Singularities	41
	2.2.3 Singularities as Silhouette Points	43
	2.2.4 Indeterminacies in the Mechanism Trajectory	47
	2.3 Lower-Level Singularity Types	48
	2.4 A Simple Mechanism with All Singularities	54
	References	56
	3 Numerical Computation of Singularity Sets	57
	3.1 A Suitable Approach	57
	3.2 Formulating the Equations of the Singularity Set	59
	3.2.1 The Assembly Constraints	59
	3.2.2 The Velocity Equation	62
	3.3 Isolating the Singularity Set	65
	3.3.1 Reduction to a Simple Quadratic Form	65
	3.3.2 Initial Bounding Box	66
	3.3.3 A Branch-and-Prune Method	66
	3.3.4 Computational Cost and Parallelization	70
	3.4 Visualising the Singularity Sets	70
	3.5 Case Studies	72
	3.5.1 The 3-RRR Mechanism	72
	3.5.2 The Gough-Stewart Platform	81
	3.5.3 A Double-Loop Mechanism	86
	References	92
	4 Workspace Determination	94
	4.1 The Need of a General Method	94
	4.2 The Workspace and Its Boundaries	95
	4.2.1 Jacobian Rank Deficiency Conditions	96
	4.2.2 Barrier Analysis	98
	4.3 Issues of Continuation Methods	100
	4.4 Exploiting the Branch-and-Prune Machinery	103
	4.4.1 Joint Limit Constraints	103
	4.4.2 Equations of the Generalised Singularity Set	105
	4.4.3 Numerical Solution and Boundary Identification	108
	4.5 Case Studies	110
	4.5.1 Multicomponent Workspaces	110
	4.5.2 Hidden Barriers	111
	4.5.3 Degenerate Barriers	117
	4.5.4 Very Complex Mechanisms	126
	References	127
	5 Singularity-Free Path Planning	130
	5.1 Related Work	131
	5.2 Formulating the Singularity-Free C-Space	132
	5.3 Exploring the Singularity-Free C-Space	135
	5.3.1 Constructing a Chart	135
	5.3.2 Constructing an Atlas	136
	5.3.3 Focusing on the Path to the Goal	140
	5.3.4 The Planner Algorithm	141
	5.4 Case Studies	143
	5.4.1 A Simple Example	144
	5.4.2 A 3-underlineRRR Parallel Robot	145
	References	153
	6 Planning with Further Constraints	156
	6.1 Wrench-Feasibility Constraints	157
	6.2 The Planning Problem	161
	6.2.1 A Characterisation of the Wrench-Feasible C-Space	161
	6.2.2 Conversion into Equality Form	163
	6.2.3 The Navigation Manifold	164
	6.2.4 Addition of Pose Constraints	165
	6.3 Proofs of the Properties	165
	6.3.1 Nonsingularity of the Screw Jacobian	166
	6.3.2 Smoothness of the Navigation Manifold	166
	6.3.3 Smoothness of Lower-Dimensional Subsets	167
	6.4 Case Studies	169
	6.4.1 Planning in Illustrative Slices	169
	6.4.2 Planning in the IRI Hexacrane	174
	6.4.3 Planning in Rigid-Limbed Hexapods	177
	6.4.4 Problem Sizes and Computation Times	178
	6.5 Details About the Wrench Ellipsoid	179
	6.6 Extensions	182
	References	183
	7 Conclusions	185
	7.1 Summary of Results	185
	7.2 Future Research Directions	187
	References	190
	Author Index	192
	Subject Index	196